Asymptotic behavior of solutions of the damped Boussinesq equation in two space dimensions

The Cauchy problem for the damped Boussinesq equation with small initial data is considered in two space dimensions. Existence and uniqueness of its classical solution is proved and the solution is constructed in the form of a series. The major term of its long-time asymptotics is calculated explicitly and a uniform in space estimate of the residual term is given

Small Wave – Vortex Disturbances in Stratified Fluid

AbstractThe theoretical description of small hydrodynamic perturbations caused by mass, force and thermal sources in some models of stratified fluid is given. The focus is on the model of a uniformly stratified heat-conducting viscous fluid. It is shown that the small perturbations can be conveniently described by several scalar quasipotentials. One quasipotential is defined by solution of the inhomogeneous differential equation of diffusion. Other quasipotentials satisfy the same high order differential equations with different right-hand sides. The linear differential operator of these equations plays a key role in the theory of small perturbations and corresponding Green's function. It is established that Green's function of small perturbations in an incompressible stratified heat-conducting viscous fluid vanishes at negative times, i.e. satisfies the causality condition. Analysis of the integral Fourier expansion of Green's function in frequencies and wave numbers is performed. It is shown that small perturbations are divided into the aperiodically damped perturbations with large wave numbers and the damped internal waves with small wave numbers. The simplifications arising in the case of unit Prandtl's number and in the limit of ideal stratified fluid are found

Analytic model for a frictional shallow-water undular bore

We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by friction. This is derived in Riemann variables using a modified finite-gap integration technique for the AKNS scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup-Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.Comment: 24 page

Thermal diffusion of solitons on anharmonic chains with long-range coupling

We extend our studies of thermal diffusion of non-topological solitons to anharmonic FPU-type chains with additional long-range couplings. The observed superdiffusive behavior in the case of nearest neighbor interaction (NNI) turns out to be the dominating mechanism for the soliton diffusion on chains with long-range interactions (LRI). Using a collective variable technique in the framework of a variational analysis for the continuum approximation of the chain, we derive a set of stochastic integro-differential equations for the collective variables (CV) soliton position and the inverse soliton width. This set can be reduced to a statistically equivalent set of Langevin-type equations for the CV, which shares the same Fokker-Planck equation. The solution of the Langevin set and the Langevin dynamics simulations of the discrete system agree well and demonstrate that the variance of the soliton increases stronger than linearly with time (superdiffusion). This result for the soliton diffusion on anharmonic chains with long-range interactions reinforces the conjecture that superdiffusion is a generic feature of non-topological solitons.Comment: 11 figure

Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system

We consider a dissipative, dispersive system of Boussinesq type, describing wave phenomena in settings where dissipation has an effect. Examples include undular bores in rivers or oceans where dissipation due to turbulence is important for their description. We show that the model system admits traveling wave solutions known as diffusive-dispersive shock waves, and we categorize them into oscillatory and regularized shock waves depending on the relationship between dispersion and dissipation. Comparison of numerically computed solutions with laboratory data suggests that undular bores are accurately described in a wide range of phase speeds. Undular bores are often described using the original Peregrine system which, even if not possessing traveling waves tends to provide accurate approximations for appropriate time scales. To explain this phenomenon, we show that the error between the solutions of the dissipative versus the non-dissipative Peregrine systems are proportional to the dissipation times the observational time